{
  "abstract": "We record lower bounds for permanent circuits with valid division over characteristic zero, counting nonscalar multiplications and divisions while additions and scalar operations are free. Chapter 5 of OpenAI's Ten Advances supplies the block construction and critical-locus estimate; these, with the parameter choice made here and the classical bounds of Strassen and Baur-Strassen, give liminf as n tends to infinity of L_div(per_n)/(n^2 log_2 log_2 n) >= 1/12. Our finite-parameter refinement for matching-minor polynomials observes that the coefficient vector at deletion size h lies in a space of dimension at most min{binom(t,h), binom(t,d-h)}. Twelve machine-checked declarations of the Lean development accompanying OpenAI's manuscript on border determinantal complexity of the permanent (September 24, 2026) imply, by a short written argument, the existence of a geometric slice; a further written, unformalized application of the same criterion gives L_div(per_n) >= (n^2/119790) log_2(n/44) for n >= 1936. Assuming Proposition 7.3 of that manuscript gives the bound (n^2/86400) log_2(n/48) for n >= 1408. These are order-of-growth results; the elementary Hessian bound n^2/2 is larger at practical orders.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.29568",
  "author_total": 1,
  "authors": [
    {
      "name": "Jizhou Guo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/egx3mk3zv8z3v.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.29568",
  "fulltext_url": null,
  "id": "egx3mk3zv8z3v",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.29568",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.AG"
    ],
    [
      "openalex",
      "1703|Complexity and Algorithms in Graphs"
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  ],
  "openalex": "W7214396425",
  "pdf_url": "https://arxiv.org/pdf/2609.29568",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/egx3mk3zv8z3v.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:58Z",
      "native_id": "2609.29568",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.29568"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "An $n^2\\log\\log n$ Lower Bound for Permanent Circuits with Valid Division",
  "title_text": "An n^2loglog n Lower Bound for Permanent Circuits with Valid Division",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:58Z",
  "url": "https://research.draconitic.com/paper/egx3mk3zv8z3v",
  "venue": "arXiv",
  "versions": []
}